
where \({\mathcal {K}}\subseteq \mathbb {R}^n\)  is a cone. The concept of set-completely positive matrices was introduced in [1]} for \({\mathcal {K}}= \mathbb {R}^n_+\) , in which case we speak of completely positive matrices. Certifying membership in \(\mathcal {CPP}(\mathbb {R}^n_+)\)  is NP-hard, so that for practical purposes, one has to work with approximations, which we will discuss shortly. In case \({\mathcal {K}}\)  is the second order cone, there is a simple characterization of \(\mathcal {CPP}\left({\mathcal {K}}\right)\)  as the intersection of the positive semidefinite matrix cone and one appropriate halfspace. In case \({\mathcal {K}}= \mathbb {R}^n\)  the respective set-completely positive cone is simply the cone of positive semidefinite matrices. For more details on set-completely positive matrices see [2]}, [3]}, [4]}.
